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Full text of "Treatise On Analysis Vol-Ii"

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for the injection JS?c (X, ju) - 3? c(X, ju) (with respect to the corresponding
seminorms) because it follows from (13.12.5) that N2(/) ^ N^/)  MX)1/2
for all/e.Sfj?(X,/0.

(13.20.5) In particular (13.9.17), if /j is a bounded (complex) measure, every
function fe%g(X) is 0-integrable, and we have I jfdfj, g HI  ||/|| by
(13.16.5). In other words, / i-> |/ d^ is a continuous linear form on the

Banach space ^c W- But it should be noted that, in general, there exist
continuous linear forms on this space which are not of this type.

(1 3.20.6) The space M(X) is also the dual of the closure ^g(X) of Jf C(X) in
the Banach space #(X) (12.15). A function / belonging to *g(X) may be
characterized by the following property :for each s > 0, there exists a compact
subset K ofX such that |/(jc)| ^ sfor allxeX-K. For if/e ?(X), then for
each & > 0 there exists by definition a function g e ^TC(X) such that
||/ #|| :gj e; if K is the support of g, then \f(x)\ ^ e for all x $ K. Conversely,
suppose that / has the above property, and let h be a continuous mapping
of Xinto [0, 1], with compact support and equal to 1 on K ((3.18.2) and
(4.5.2)); it is clear that \\f-fh\\  s and that//ze Jfc(X), hence /e g(X).
The functions belonging to ^cPO are called (complex-valued) continuous
functions which tend to 0 at infinity. (When X = R, they are indeed the
continuous functions / such that lim /(#) = lim f(x) = 0.) We put

n ^R(X) for the corresponding space of real-valued functions.
When X is compact, we have


1.   The space Mi(X) of bounded real measures on a locally compact space X can be
considered as a space of linear forms on each of the following vector spaces :

(1)    the space EI = ^TR(X) of continuous functions with compact support;

(2)    the space E2 = #a(X) of continuous functions which tend to 0 at infinity;

(3)    the space E3 = ^f(X) of bounded continuous functions;

(4)    the space E5 of linear combinations of (upper or lower) semicontinuous
bounded functions;

(5)    the space E6 = ^n(X) of universally measurable bounded functions.

Moreover, if v is any positive measure on X, the space Mi(X, v) of bounded
measures with base v (which may be identified with the space L(X, v) by virtue of
13.14.4)) can be considered as a space of linear forms on the vector space E4,v of
bounded functions which are continuous almost everywhere with respect to v. y fn is everywhere finite and continuous (because every point